Review – First Test on Derivatives
[1] Find f   x  .
2
[a] f  x   4 x  5 x  3
[e] f  x  
x3
2x 1
[i] f  x   ln  3x  1
1
x
2
[b] f  x   16
[c] f  x  
[f] f  x   x sin x
2 x
[g] f  x   x e
[j] f  x  
4
x5
sin x
[h] f  x   2 x
e
[d] f  x  
 
ln x
x
3x
[k] f  x   ln e
[l] f  x   ln  tan x 
4
[2] Write an equation for the line tangent to the graph of y  x  3x 2 at x  2 .
[3] Write an equation for the line tangent to the graph of y  e 2 x1 at x  1 .
[4] Write an equation for the line normal to the graph of y  ln x at x  e 4 .
[5] For each function, find the interval on which each it is increasing and state where it has extrema.
2
[a] f  x   x  x
3
2
[b] f  x   x  3x
[c] f  x  
x
ex
[6] For each function, find the interval on which each it is concave up and state where it has a flex point.
3
2
[a] f  x   x  3x
[b] f  x  
x
ex
1
3
[7] A particle is moving along the y-axis. Its displacement from the origin is given by s  t   t 3  3t 2  8t  5 , where t
is in seconds and s is in meters.
[a] Where is the particle at time t  1 ?
[b] Find the instantaneous velocity of the particle at time t  1 .
[c] Find its average velocity on the time interval 0  t  2 .
2
[8] The height of a ball thrown vertically upward is given by h  t   4.9t  4t  1 , where t is in seconds and h is in
meters.
[a] The height is thrown initially from what height?
[b] Find the maximum height of the ball.
[c] After how many seconds does the ball hit the ground ( h  0 )?